Periodic Functions - UniMAP Portalportal.unimap.edu.my/portal/page/portal30/Lecturer Notes...The...
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Fourier Series Periodic Functions
Transcript of Periodic Functions - UniMAP Portalportal.unimap.edu.my/portal/page/portal30/Lecturer Notes...The...
Fourier Series
Periodic Functions
The Mathematic Formulation
Any function that satisfies
( ) ( )f t f t T
where T is a constant and is called the period
of the function.
Example:
4cos
3cos)(
tttf Find its period.
)()( Ttftf )(4
1cos)(
3
1cos
4cos
3cos TtTt
tt
Fact: )2cos(cos m
mT
23
nT
24
mT 6
nT 8
24T smallest T
Example:
tttf 21 coscos)( Find its period.
)()( Ttftf )(cos)(coscoscos 2121 TtTttt
mT 21
nT 22
n
m
2
1
2
1
must be a
rational number
Example:
tttf )10cos(10cos)(
Is this function a periodic one?
10
10
2
1 not a rational
number
Fourier Series
Fourier Series
Introduction
Decompose a periodic input signal into
primitive periodic components.
A periodic sequence
T 2T 3T
t
f(t)
Synthesis
T
ntb
T
nta
atf
n
n
n
n
2sin
2cos
2)(
11
0
DC Part Even Part Odd Part
T is a period of all the above signals
)sin()cos(2
)( 0
1
0
1
0 tnbtnaa
tfn
n
n
n
Let 0=2/T.
Orthogonal Functions
Call a set of functions {k} orthogonal
on an interval a < t < b if it satisfies
nmr
nmdttt
n
b
anm
0)()(
Decomposition
dttfT
aTt
t
0
0
)(2
0
,2,1 cos)(2
0
0
0
ntdtntfT
aTt
tn
,2,1 sin)(2
0
0
0
ntdtntfT
bTt
tn
)sin()cos(2
)( 0
1
0
1
0 tnbtnaa
tfn
n
n
n
Example (Square Wave)
112
2
00
dta
,2,1 0sin1
cos2
200
nntn
ntdtan
,6,4,20
,5,3,1/2)1cos(
1 cos
1sin
2
200
n
nnn
nnt
nntdtbn
2 3 4 5--2-3-4-5-6
f(t)1
112
2
00
dta
,2,1 0sin1
cos2
200
nntn
ntdtan
,6,4,20
,5,3,1/2)1cos(
1 cos
1sin
2
100
n
nnn
nnt
nntdtbn
2 3 4 5--2-3-4-5-6
f(t)1
Example (Square Wave)
ttttf 5sin
5
13sin
3
1sin
2
2
1)(
112
2
00
dta
,2,1 0sin1
cos2
200
nntn
ntdtan
,6,4,20
,5,3,1/2)1cos(
1 cos
1sin
2
100
n
nnn
nnt
nntdtbn
2 3 4 5--2-3-4-5-6
f(t)1
Example (Square Wave)
-0.5
0
0.5
1
1.5
ttttf 5sin
5
13sin
3
1sin
2
2
1)(
Harmonics
T
ntb
T
nta
atf
n
n
n
n
2sin
2cos
2)(
11
0
DC Part Even Part Odd Part
T is a period of all the above signals
)sin()cos(2
)( 0
1
0
1
0 tnbtnaa
tfn
n
n
n
Harmonics
tnbtnaa
tfn
n
n
n 0
1
0
1
0 sincos2
)(
Tf
22 00
Define , called the fundamental angular frequency.
0 nnDefine , called the n-th harmonic of the periodic function.
tbtaa
tf n
n
nn
n
n
sincos2
)(11
0
Harmonics
tbtaa
tf n
n
nn
n
n
sincos2
)(11
0
)sincos(2 1
0 tbtaa
nnn
n
n
12222
220 sincos2 n
n
nn
nn
nn
nnn t
ba
bt
ba
aba
a
1
220 sinsincoscos2 n
nnnnnn ttbaa
)cos(1
0 n
n
nn tCC
Amplitudes and Phase Angles
)cos()(1
0 n
n
nn tCCtf
2
00
aC
22
nnn baC
n
nn
a
b1tan
harmonic amplitude phase angle
Fourier Series
Complex Form of the Fourier Series
Complex Exponentials
tnjtnetjn
00 sincos0
tjntjneetn 00
2
1cos 0
tnjtnetjn
00 sincos0
tjntjntjntjnee
jee
jtn 0000
22
1sin 0
Complex Form of the Fourier Series
tnbtnaa
tfn
n
n
n 0
1
0
1
0 sincos2
)(
tjntjn
n
n
tjntjn
n
n eebj
eeaa
0000
11
0
22
1
2
1
0 00 )(2
1)(
2
1
2 n
tjn
nn
tjn
nn ejbaejbaa
1
000
n
tjn
n
tjn
n ececc
)(2
1
)(2
1
2
00
nnn
nnn
jbac
jbac
ac
Complex Form of the Fourier Series
1
000)(
n
tjn
n
tjn
n ececctf
1
1
000
n
tjn
n
n
tjn
n ececc
n
tjn
nec 0
)(2
1
)(2
1
2
00
nnn
nnn
jbac
jbac
ac
Complex Form of the Fourier Series
2/
2/
00 )(
1
2
T
Tdttf
T
ac
)(2
1nnn jbac
2/
2/0
2/
2/0 sin)(cos)(
1 T
T
T
Ttdtntfjtdtntf
T
2/
2/00 )sin)(cos(
1 T
Tdttnjtntf
T
2/
2/
0)(1 T
T
tjndtetf
T
2/
2/
0)(1
)(2
1 T
T
tjn
nnn dtetfT
jbac )(2
1
)(2
1
2
00
nnn
nnn
jbac
jbac
ac
Complex Form of the Fourier Series
n
tjn
nectf 0)(
dtetfT
cT
T
tjn
n
2/
2/
0)(1
)(2
1
)(2
1
2
00
nnn
nnn
jbac
jbac
ac
If f(t) is real,
*
nn cc
nn j
nnn
j
nn ecccecc
|| ,|| *
22
2
1|||| nnnn bacc
n
nn
a
b1tan
,3,2,1 n
002
1ac
Complex Frequency Spectra
nn j
nnn
j
nn ecccecc
|| ,|| *
22
2
1|||| nnnn bacc
n
nn
a
b1tan ,3,2,1 n
002
1ac
|cn|
amplitude
spectrum
n
phase
spectrum
Example
2
T
2
T TT
2
d
t
f(t)
A
2
d
dteT
Ac
d
d
tjn
n
2/
2/
0
2/
2/0
01
d
d
tjne
jnT
A
2/
0
2/
0
0011 djndjn
ejn
ejnT
A
)2/sin2(1
0
0
dnjjnT
A
2/sin1
0
021
dnnT
A
T
dn
T
dn
T
Adsin
T
dn
T
dn
T
Adcn
sin
82
5
1
T ,
4
1 ,
20
1
0
T
dTd
Example
40 80 120-40 0-120 -80
A/5
50 100 150-50-100-150
T
dn
T
dn
T
Adcn
sin
42
5
1
T ,
2
1 ,
20
1
0
T
dTd
Example
40 80 120-40 0-120 -80
A/10
100 200 300-100-200-300
Example
dteT
Ac
dtjn
n
0
0
d
tjne
jnT
A
00
01
00
110
jne
jnT
A djn
)1(1
0
0
djne
jnT
A
2/0
sindjn
e
T
dn
T
dn
T
Ad
TT d
t
f(t)
A
0
)(1 2/2/2/
0
000 djndjndjneee
jnT
A
Fourier Series
Impulse Train
Dirac Delta Function
0
00)(
t
tt and 1)(
dtt
0t
Also called unit impulse function.
Property
)0()()(
dttt
)0()()0()0()()()(
dttdttdttt
(t): Test Function
Impulse Train
0t
T 2T 3TT2T3T
n
T nTtt )()(
Fourier Series of the Impulse Train
n
T nTtt )()(T
dttT
aT
TT
2)(
2 2/
2/0
Tdttnt
Ta
T
TTn
2)cos()(
2 2/
2/0
0)sin()(2 2/
2/0 dttnt
Tb
T
TTn
n
T tnTT
t 0cos21
)(
Complex FormFourier Series of the Impulse Train
Tdtt
T
ac
T
TT
1)(
1
2
2/
2/
00
Tdtet
Tc
T
T
tjn
Tn
1)(
1 2/
2/
0
n
tjn
T eT
t 01
)(
n
T nTtt )()(
